I've got a problem that I'm struggling to put into a form that I can analyze.
Suppose I have a quadratic form $f(x,y)=ax^2+2bxy+cy^2 = \mathbf{u}\mathbf{A}\mathbf{u}^T$ for $\mathbf{u} = \begin{bmatrix}x&y\end{bmatrix}$, $\mathbf{A} = \begin{bmatrix}a & b \cr b &c\end{bmatrix}$
Now I have two series of coordinate pairs $\mathbf{u}_k = \begin{bmatrix}x_k&y_k\end{bmatrix}$ and $\mathbf{v}_k = \begin{bmatrix}x_k&y_k\end{bmatrix}$, and I want to compare the sets $U=\{\mathbf{u}_k\}$ and $V=\{\mathbf{v}_k\}$ by evaluating the metrics
$$g(U) = \max\limits_{\lVert{A}\rVert \le 1} \sum\limits_k f(x_k,y_k)$$
and comparing $g(U)$ and $g(V)$.
Is there an easy way to compute $g(U)$ from knowing the individual $(x_k,y_k)$ pairs? I can compute the following quantities easily:
$$\begin{align}
S_{xx} &= \sum x^2 \cr
S_{xy} &= \sum xy \cr
S_{yy} &= \sum y^2
\end{align}$$
I just am not sure how to use those to compute the maximum over the constraint $\lVert{A}\rVert \le 1$. (I have a very poor knowledge of matrix norm identities.)
(Important fact: I'm using Frobenius norms.)
Given the Frobenius norm definition, this means that I'm looking for the maximum value of $aS_{xx} + 2bS_{xy} + cS_{yy}$ subject to the constraint $a^2 + 2b^2 + c^2 \le 1$.
Not sure where to go from here, though it seems like I'm really close.
Best Answer
Using method of Lagrange multipliers you can find that
$$ \underset{a^2 + 2b^2 + c^2 \le 1}{\operatorname{max}} \Big(aS_{xx} + 2bS_{xy} + cS_{yy}\Big)=\sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}, $$
which is obtained when
$$ a=\frac{S_{xx}}{\sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}}, \quad b=\frac{S_{xy}}{\sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}}, \quad c=\frac{S_{yy}}{\sqrt{S_{xx}^2 + 2S_{xy}^2 + S_{yy}^2}} $$